CS4961 Parallel Programming
Lecture 2:
Introduction to Parallel
Algorithms
Mary Hall
August 26, 2010
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Homework 1 – Due 10:00 PM, Wed., Sept. 1
• To submit your homework:
- Submit a PDF file
- Use the “handin” program on the CADE machines
- Use the following command:
“handin cs4961 hw1 <prob1file>”
Problem 1:
- What are your goals after this year and how do you
anticipate this class is going to help you with that? Some
possible answers, but please feel free to add to them. Also,
please write at least one sentence of explanation.
- A job in the computing industry
- A job in some other industry where computing is applied to
real-world problems
- As preparation for graduate studies
- Intellectual curiosity about what is happening in the computing
field
- Other
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Homework 1
Problem 2:
(a) Provide pseudocode (as in the book and class notes) for a
correct and efficient parallel implementation in C of the
parallel prefix computation (see Fig. 1.4 on page 14).
Assume your input is a vector of n integers, and there are
n/2 processors. Each processor is executing the same
thread code, and the thread index is used to determine
which portion of the vector the thread operates upon and
the control. For now, you can assume that at each step in
the tree, the threads are synchronized.
(b) The structure of this and the tree-based parallel sums
from Figure 1.3 are similar to the parallel sort we did with
the playing cards on Aug. 24. In words, describe how you
would modify your solution of (a) above to derive the
parallel sorting implementation.
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Homework 1, cont.
Problem 3 (see supplemental notes for definitions):
Loop reversal is a transformation that reverses the
order of the iterations of a loop. In other words,
loop reversal transforms a loop with header
for (i=0; i<N; i++)
into a loop with the same body but header
for (i=N-1; i>=0; i--)
Is loop reversal a valid reordering transformation
on the “i” loop in the following loop nest? Why or
why not?
for (j=0; j<N; j++)
for (i=0; i<N; i++)
a[i+1][j+1] = a[i][j] + c;
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Today’s Lecture
• Parallelism in Everyday Life
• Learning to Think in Parallel
• Aspects of parallel algorithms (and a hint at
complexity!)
• Derive parallel algorithms
• Discussion
• Sources for this lecture:
- Larry Snyder,
“http://www.cs.washington.edu/education/courses
/524/08wi/”
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Is it really harder to “think” in parallel?
• Some would argue it is more natural to think
in parallel…
• … and many examples exist in daily life
• Examples?
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Is it really harder to “think” in parallel?
• Some would argue it is more natural to think
in parallel…
• … and many examples exist in daily life
- House construction -- parallel tasks, wiring and
plumbing performed at once (independence), but
framing must precede wiring (dependence)
- Similarly, developing large software systems
- Assembly line manufacture - pipelining, many
instances in process at once
- Call center - independent calls executed
simultaneously (data parallel)
- “Multi-tasking” – all sorts of variations
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Reasoning about a Parallel Algorithm
• Ignore architectural details for now
• Assume we are starting with a sequential
algorithm and trying to modify it to execute in
parallel
- Not always the best strategy, as sometimes the
best parallel algorithms are NOTHING like their
sequential counterparts
- But useful since you are accustomed to sequential
algorithms
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Reasoning about a parallel algorithm, cont.
• Computation Decomposition
- How to divide the sequential computation among
parallel threads/processors/computations?
• Aside: Also, Data Partitioning (ignore today)
• Preserving Dependences
- Keeping the data values consistent with respect
to the sequential execution.
• Overhead
- We’ll talk about some different kinds of
overhead
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Race Condition or Data Dependence
• A race condition exists when the result of an
execution depends on the timing of two or
more events.
• A data dependence is an ordering on a pair of
memory operations that must be preserved to
maintain correctness.
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Key Control Concept: Data Dependence
• Question: When is parallelization guaranteed to be
safe?
• Answer: If there are no data dependences across
reordered computations.
• Definition: Two memory accesses are involved in a
data dependence if they may refer to the same
memory location and one of the accesses is a write.
• Bernstein’s conditions (1966): Ij is the set of
memory locations read by process Pj, and Oj the set
updated by process Pj. To execute Pj and another
process Pk in parallel,
Ij ∩ Ok = ϕ
write after read
Ik ∩ Oj = ϕ
read after write
Oj ∩ Ok = ϕ
write after write
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Data Dependence and Related Definitions
• Actually, parallelizing compilers must formalize this to guarantee
correct code.
• Let’s look at how they do it. It will help us understand how to reason
about correctness as programmers.
• Definition:
Two memory accesses are involved in a data dependence if they may
refer to the same memory location and one of the references is a
write.
A data dependence can either be between two distinct program
statements or two different dynamic executions of the same program
statement.
• Source:
• “Optimizing Compilers for Modern Architectures: A Dependence-Based
Approach”, Allen and Kennedy, 2002, Ch. 2. (not required or essential)
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Data Dependence of Scalar Variables
True (flow) dependence
a
Anti-dependence
a
Output dependence
a
a
=
=a
=a
=
=
=
Input dependence (for locality)
=a
=a
Definition: Data dependence exists from a reference
instance i to i’ iff
either i or i’ is a write operation
i and i’ refer to the same variable
i executes before i’
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Some Definitions (from Allen & Kennedy)
• Definition 2.5:
- Two computations are equivalent if, on the same inputs,
- they produce identical outputs
- the outputs are executed in the same order
• Definition 2.6:
- A reordering transformation
- changes the order of statement execution
- without adding or deleting any statement executions.
• Definition 2.7:
- A reordering transformation preserves a dependence if
- it preserves the relative execution order of the dependences’
source and sink.
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Fundamental Theorem of Dependence
• Theorem 2.2:
- Any reordering transformation that preserves
every dependence in a program preserves the
meaning of that program.
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Simple Example 1:
“Hello World” of Parallel Programming
• Count the 3s in array[] of length values
• Definitional solution … Sequential program
count = 0;
for (i=0; i<length; i++) {
if (array[i] == 3)
count += 1;
}
Can we rewrite this to a parallel code?
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Computation Partitioning
• Block decomposition: Partition original loop
into separate “blocks” of loop iterations.
- Each “block” is assigned to an independent
“thread” in t0, t1, t2, t3 for t=4 threads
- Length = 16 in this example
3
0
7
2
3
3
1
0
2
0
9
1
3
2
2
3
1
0
{
{
{
{
2
t0
t1
t2
t3
int block_length_per_thread = length/t;
Correct?
int start = id * block_length_per_thread;
Preserve
for (i=start; i<start+block_length_per_thread; i++) {
Dependences?
if (array[i] == 3)
count += 1;
}
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Data Race on Count Variable
• Two threads may interfere on memory writes
3
0
7
2
3
3
1
0
2
0
9
1
3
2
2
3
1
0
{
{
{
{
2
t0
t1
t2
Thread 1
load count
Thread 3
count = 0
count = 1
count = 2
increment count
store count
load count
increment count
count = 1
store<count,1>
store<count,2>
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What Happened?
• Dependence on count across
iterations/threads
- But reordering ok since operations on count are
associative
• Load/increment/store must be done
atomically to preserve sequential meaning
• Definitions:
- Atomicity: a set of operations is atomic if either
they all execute or none executes. Thus, there
is no way to see the results of a partial
execution.
- Mutual exclusion: at most one thread can
execute the code at any time
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Try 2: Adding Locks
• Insert mutual exclusion (mutex) so that only
one thread at a time is
loading/incrementing/storing count
atomically
int block_length_per_thread = length/t;
mutex m;
int start = id * block_length_per_thread;
for (i=start; i<start+block_length_per_thread; i++) {
if (array[i] == 3) {
mutex_lock(m);
count += 1;
mutex_unlock(m);
}
}
Correct now. Done?
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Performance Problems
• Serialization at the mutex
• Insufficient parallelism granularity
• Impact of memory system
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Lock Contention and Poor Granularity
• To acquire lock, must go
through at least a few levels of
cache (locality)
• Local copy in register not going to be
correct
• Not a lot of parallel work
outside of acquiring/releasing
lock
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Try 3: Increase “Granularity”
• Each thread operates on a private copy of count
• Lock only to update global data from private
copy
mutex m;
int block_length_per_thread = length/t;
int start = id * block_length_per_thread;
for (i=start; i<start+block_length_per_thread; i++) {
if (array[i] == 3)
private_count[id] += 1;
}
mutex_lock(m);
count += private_count[id];
mutex_unlock(m);
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Much Better, But Not Better than Sequential
• Subtle cache effects are limiting performance
Private variable ≠
Private cache line
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Try 4: Force Private Variables into
Different Cache Lines
• Simple way to do this?
• See textbook for authors’ solution
Parallel speedup when <t = 2>:
time(1)/time(2) = 0.91/0.51
= 1.78 (close to number of processors!)
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Discussion: Overheads
• What were the overheads we saw with this
example?
- Extra code to determine portion of computation
- Locking overhead: inherent cost plus contention
- Cache effects: false sharing
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Generalizing from this example
• Interestingly, this code represents a common pattern
in parallel algorithms
• A reduction computation
- From a large amount of input data, compute a smaller result
that represents a reduction in the dimensionality of the input
- In this case, a reduction from an array input to a scalar result
(the count)
• Reduction computations exhibit dependences that must
be preserved
- Looks like “result = result op …”
- Operation op must be associative so that it is safe to reorder
them
• Aside: Floating point arithmetic is not truly associative,
but usually ok to reorder
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Simple Example 2:
Another “Hello World” Equivalent
• Parallel Summation:
- Adding a sequence of numbers A[0],…,A[n-1]
• Standard way to express it
sum = 0;
for (i=0; i<n; i++) {
sum += A[i];
}
• Semantics require:
(…((sum+A[0])+A[1])+…)+A[n-1]
• That is, sequential
• Can it be executed in parallel?
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Graphical Depiction of Sum Code
Original Order
Pairwise Order
Which decomposition is better suited for parallel execution.
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Summary of Lecture
• How to Derive Parallel Versions of Sequential Algorithms
- Computation Partitioning
- Preserving Dependences and Reordering
Transformations
- Reduction Computations
- Overheads
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Next Time
• A Discussion of parallel computing platforms
• Questions about first written homework assignment
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